In the previous topic we discussed about the method of getting the sum of finite sine series. Now we will extend our learning to cosines. In this topic I will give a three simple and precise method how to deal with this kind of problem.
- Express the series in terms of summation expression
- Determine the value of the following;
- n – number of terms which can be determined by using the sum of arithmetic sequence
- x – this is the multiplier
- Use the formula provided and simplify the expression by applying the sum or difference of angle formula.
Sample Problem 1:
Express the sum of cos1° + cos2° + . . . + cos90° and give the sum as a single trigonometric expression.
Following the process mentioned above
a. We can express the expression this way,
b. Clearly, n=90 because there are 90 terms. Also it is obvious that x=1.
c. Using the formula:
Using addition formula we have,
Going back to our formula,
Sample Problem 2:
cos2+cos4+ cos6 + . . . +cos60 can be expressed as . Where x,y,z are integers. What is the value of A+B+C?
To solve this we still use the same process
The equivalent summation expression for this is
n=30 because there are only 30 terms in the series. x=2.
sin30=1/2 and cos31=cos(30+1)
By addition formula of cosine:
Going back to our formula:
Now your turn
Without using your calculator,find the sum of cos4+cos8+cos12+. . .+cos180